Exactly one of these four equations has no real roots. Which one is it?

Class 10Quadratic EquationsAlgebra — Quadratics

Step-by-Step Solution

Compute

D

for each.

4 x^{2} - 3 x + 2 : D = 9 - 32 = - 23 < 0

, so no real roots.

3 x^{2} - 4 x + 1 : D = 16 - 12 = 4 > 0

(two real roots).

x^{2} - 2 x - 5 : D = 4 + 20 = 24 > 0

(two real roots — note that with

c

negative,

- 4 a c

ADDS to

b^{2}

, so

D

can never be negative).

9 x^{2} - 12 x + 4 : D = 144 - 144 = 0

, which gives equal real roots, not "no roots".

Answer: C)

4 x^{2} - 3 x + 2 = 0

D = b^{2} - 4 a c

classifies the roots of

a x^{2} + b x + c = 0 : - * * D > 0

**: TWO distinct real roots. Parabola crosses

x

-axis at two points.
- **

D = 0

**: ONE repeated real root (called a 'double root' or 'two equal roots'). Parabola touches

x

-axis at the vertex.
- **

D < 0

**: NO real roots. Parabola does not touch the

x

-axis at all.

**Use case**: 'For what values of

k

does

k x^{2} + 2 x + 1 = 0

have equal roots?' → set

D = 0 : 4 - 4 k = 0

→

k = 1

.

You can answer 'how many real roots?' WITHOUT computing them — just compute

D

Don't just memorise the answer — build a daily practice habit.